(CNN) -- Just as it was four years earlier, the 2004 presidential vote couldn't be settled on Election Night.

Sen. John Kerry's campaign refused to concede early Wednesday, arguing that more than 250,000 provisional and absentee ballots remained to be counted in the key battleground state of Ohio.

New Mexico will not release presidential election results until later Wednesday because thousands of absentee ballots remain uncounted, according to a spokesman for the secretary of state.

The vote also remained unsettled in Iowa, where broken machines, a delay in opening absentee ballots and a delay in the arrival of other absentee ballots were slowing completion of the secretary of state's final count in the presidential election, elections officials said early Wednesday.

But Kerry's presidential hopes appeared to hang on the outcome of the vote in Ohio.

"We've waited four years for this victory," Kerry's running mate, Sen. John Edwards, told supporters in Boston. "We can wait one more night."

With 97 percent of precincts reporting, Kerry trailed President Bush in Ohio by about 125,000 votes out of more than 5.2 million cast. But in a statement early Wednesday, Kerry campaign chief Mary Beth Cahill said, "The vote count in Ohio has not been completed."

"There are more than 250,000 remaining votes to be counted," Cahill said. "We believe when they are, John Kerry will win Ohio."

Ohio Secretary of State Ken Blackwell issued orders for counties by 2 p.m. Wednesday to report total numbers of provisional ballots. Counting of those ballots will not begin until Thursday, according to Blackwell's directive. He initially said the counting of provisional and absentee ballots would not begin for 11 days.

"This is a very deliberate and cautious process," he told CNN. "And so, you know, I tell everybody just take a deep breath and relax. We can't predict what the results are going to be. We can only guarantee you you're going to get an honest and fair count through our bipartisan system."

Ohio's two U.S. senators, both Republicans, issued a statement through the Bush campaign calling on Kerry to concede the state.

"Based on our experience in Ohio politics, we believe that the president's lead in Ohio is clear and that it cannot be surmounted," Sens. Mike DeWine and George Voinovich said. "Sen. Kerry should concede defeat and spare the country the turmoil of another drawn-out election."

But Blackwell, a Republican, told reporters at a 2 a.m. news conference that Ohio has used provisional ballots for a decade, and election officials know how to handle them.

"The system can handle a close election," he said. "It means that you might not know immediately -- but when you know, you will know that you have a result that is worthy of the full confidence of the folks who participated in the process today."

With 20 electoral votes, Ohio was among the most hotly contested states in Tuesday's election. Neither candidate appeared likely to claim the 270 electoral votes needed to win the presidency while its results remained up in the air.

While Blackwell said the exact number of provisional ballots was unknown, he said it is "trending toward 175,000."

"We will not know, and nobody knows how many provisional ballots we have had cast until all of the tabulations have come in from across the state," he said.

On absentee ballots, Blackwell spokesman Carlo LoParo said the secretary of state's office will not know how many were returned until after the election, when county election officials send in their numbers.

A federal judge Tuesday ruled that Ohio voters who had requested but had not received absentee ballots must be allowed to cast provisional ballots at the polls. But District Court Judge David Katz did not order that the state must count the provisional ballots, leaving that to be decided at a later date.

Katz's ruling required the state to provide a provisional ballot to anyone who requested one.

Blackwell has directed county boards of elections to count only those provisional ballots that were cast in the correct precinct, and not those from voters who cast them outside of their home precinct.

The open question, said CNN legal analyst Jeffrey Toobin, is whether the Kerry team will also ask for a recount of all ballots. Since most were punch cards, he said, we could be looking at chads again.

Iowa hit by vote-count woes

In Iowa, two central-count machines -- used to read paper ballots by optical scanners in Greene and Harrison counties -- broke down, said Charles Krogmeier, deputy secretary of state.

The machine in Harrison County was repaired, but the machine in Greene had to await repair or replacement by its maker, Elections Systems & Software, in Omaha, Nebraska. Results were expected by 8 a.m. (9 a.m. ET).

Harrison County tends to skew Republican and has 9,796 voters.

In addition, Lee County -- in the southeast corner of the state -- was still opening its 6,800 absentee ballots early Wednesday, said Krogmeier. So too was Montgomery County, which had 517 absentee ballots, he said.

Both counties were expected to complete their work overnight, he said.

Another wild card is the 60,000 absentee ballots that were requested but have not yet been returned, he said.

As long as those absentee ballots were postmarked on or before November 1 and are received by noon Monday, November 8, they will be counted, he said.

And about 10,000 provisional ballots will be counted Thursday, said Secretary of State Chet Culver.

"We have to be patient and we have to let the process work," he said.

Florida official: Absentee count may last days

Florida voters cast an "incredible number" of absentee ballots in Tuesday's presidential election, and counting them may take until Thursday, Secretary of State Glenda Hood said Tuesday night.

Hood said only about 30,000 of the 94,000 absentee ballots received in heavily Democratic Miami-Dade County had been counted, and officials were pulling in "extra staff" to complete the count by noon Thursday, the deadline by which counties must complete the unofficial canvass of all votes.

Broward County, north of Miami and also heavily Democratic, had 92,000 absentee ballots requested, Hood said, but the count of the ballots that were returned may be completed overnight.

Palm Beach County began counting its absentee ballots Friday. In that county, 70,000 absentee ballots were received, Hood said.

Hood said workers will continue tallying the votes around the clock until the process is completed.

With about 99 percent of the precincts reporting in Florida's Senate race, former Bush administration Housing and Urban Affairs Secretary Mel Martinez, a Republican, held a nearly 72,000-vote lead over Betty Castor, for the seat being vacated by retiring Democratic Sen. Bob Graham. But Castor refused to concede, citing uncounted votes.